Data Types as Objects

In this paper we present a new approach to the semantics of data types, in which the types themselves are incorporated as elements of the domain of data objects. The approach allows types to have subtypes, allows genuinely polymorphic functions, and gives a precise semantics for recursive type definitions (including definitions with parameters). In addition, the approach yields simple and straightforward methods for proving type properties of recursive definitions. These methods include a new fixedpoint rule which permits case analysis. 1 Informal Introduction Most current type systems (e.g., that of LCF, as in [2]) are based on [1], Church’s 1940 paper on a simple theory of types. In these systems there are a number of unrelated ground data types and an arrow operation for forming function types of finite level. A serious drawback of such systems is the requirement that each data object must have a unique type. Thus integers, for example, cannot be considered as a special case of real numbers, and in general one type cannot be a subtype of another. A second problem, related to the first, is that polymorphism is not possible; each argument of a function must be of some specified single type. For example, we cannot have a single addition operation capable of adding both reals and integers; instead, we need four functions of various types for the various possible combinations. Especially serious is the lack of a general if–then–else conditional whose domains are left unspecified. The third problem is that in a Church system there are in fact two ‘meta’-types, namely types and data objects, which cannot be mixed (LCF has variables of both metatypes). For example, the natural identity evenInteger + 1 = oddInteger relating the types evenInteger and oddInteger and the data object 1 makes no sense in a Church system (except, perhaps, informally). Even unmixed recursive type equations are problematic because the types do not constitute a domain. Our approach is to incorporate all the objects and data types together in a single unified domain. Any element of the resulting domain then has two rôles: 1. it is a data object, at which functions can be defined, including of course functions which are least fixedpoints of recursive definitions; 2. it is the type of all objects which approximate it; the assertions “x v y”, “x is of type y” and “any object of type x is of type y” are therefore all equivalent. Given an initial domain D of data objects, and a collection of intuitive data types, we form a new domain D̂ (which we will call a type extension of D) by adding the types as new data objects in the sense of (1). The extended v relation on D̂ is determined by the equivalence in part (2): ∗Published in Fourth Colloquium on Automata, Languages and Programming, LNCS 52:465–479, held in Turku, Finland on 18–22 July 1977. Preliminary version published as University of Warwick Computer Science Research Report CS–RR–020, June 1977.